Find a monic cubic polynomial P(x) with integer coefficients such that P(3√2+1)=0. (A polynomial is monic if its leading coefficient is 1.)
P(x)=x3+bx2+cx+d(21/3+1)3+b(21/3+1)2+c(21/3+1)+d=03+3(22/3+21/3)+b(22/3)+2b(21/3)+b+c(21/3)+c+d=0(3+b)(22/3)+(3+2b+c)(21/3)+(3+b+c+d)=03+b=0b=−33+2b+c=03−6+c=0c=33+b+c+d=03−3+3+d=0d=−3∴The required monic polynomial is x3−3x2+3x−3=0
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